A Cost-Scaling Algorithm for Minimum-Cost Node-Capacitated Multiflow Problem
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In this paper, we address the minimum-cost node-capacitated multiflow problem in an undirected network. For this problem, Babenko and Karzanov (2012) showed strongly polynomial-time solvability via the ellipsoid method. Our result is the first combinatorial weakly polynomial-time algorithm for this problem. Our algorithm finds a half-integral minimum-cost maximum multiflow in O(m log(nCD)SF(kn, m, k)) time, where n is the number of nodes, m is the number of edges, k is the number of terminals, C is the maximum node capacity, D is the maximum edge cost, and SF(n', m', η) is the time complexity of solving the submodular flow problem in a network of n' nodes, m' edges, and a submodular function with η-time-computable exchange capacity. Our algorithm is built on discrete convex analysis on graph structures and the concept of reducible bisubmodular flows.
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